On Newton-type methods with cubic convergence

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Journal of Computational and Applied Mathematics 176 (2005) 425 432 www.elsevier.com/locate/cam On Newton-type methods with cubic convergence H.

1 Journal of Computational and Applied Mathematics 176 (2005) On Newton-type methods with cubic convergence H.H.H. Homeier a,b, a Science + Computing Ag, IT Services Muenchen, Ingolstädter Str. 22, D80807 München, Germany b Institut für Physikalische und Theoretische Chemie, Universität Regensburg, Regenburg, Germany Received 21 June 2004 Abstract Recently, there has been some progress on Newton-type methods with cubic convergence that do not require the computation of second derivatives. Weerakoon and Fernando (Appl. Math. Lett. 13 (2000) 87) derived the Newton method and a cubically convergent variant by rectangular and trapezoidal approximations to Newton s theorem, while Frontini and Sormani (J. Comput. Appl. Math. 156 (2003) 345; 140 (2003) 419 derived further cubically convergent variants by using different approximations to Newton s theorem. Homeier (J. Comput. Appl. Math. 157 (2003) 227; 169 (2004) 161) independently derived one of the latter variants and extended it to the multivariate case. Here, we show that one can modify the Werrakoon Fernando approach by using Newton s theorem for the inverse function and derive a newclass of cubically convergent Newton-type methods Elsevier B.V. All rights reserved. MSC: 41A25; 65D99 Keywords: Rootfinding; Newton method; Newton-type method; Newton theorem; Inverse function; Iterative methods; Nonlinear equations Let f : R R be a smooth nonlinear function with a simple root x, i.e., f(x ) = 0 and f (x ) = 0. We consider iterative methods for the calculation of x that uses f and f but not the higher derivatives of f and that generalize the Newton method. Modifications for multiple roots will not be considered in the present contribution. Corresponding author. Science + Computing Ag, IT Services Muenchen, Ingolstädter Str. 22, D80807 München, Germany. Tel.: ; fax: address: herbert.homeier@na-net.ornl.gov (H.H.H. Homeier) /$ - see front matter 2004 Elsevier B.V. All rights reserved. doi: /j.cam

2 426 H.H.H. Homeier / Journal of Computational and Applied Mathematics 176 (2005) The famous Newton method for finding x uses the iterative scheme +1 = f() f (1) ( ) starting from some initial value x 0. It converges quadratically in some neighborhood of x for a simple roots x. There exists an extension due to Potra and Pták called the two-step method [1] that may be rewritten as the iterative scheme +1 = f() + f( f( )/f ( )) f (2) ( ) that converges cubically in some neighborhood of x. For quite a long time, this was the only known method converging cubically apart from methods that involve higher-order derivatives (for a recent reviewof the latter methods, see [1]). Then, Weerakoon and Fernando [8] rederived the Newton method from the Newton theorem f(x)= f( ) + f (t) dt by approximating the integral by the rectangular rule according to f (t) dt (x )f ( ) (4) and using f(x)= 0. When they used the trapezoidal approximation f (t) dt (x )(f ( ) + f (x))/2 (5) in combination with the approximation f (x) f ( f( )/f ( )) and f(x)= 0, they arrived at the modified Newton-type iterative scheme 2f( ) +1 = f ( ) + f ( f( )/f (6) ( )) and proved that this scheme converges cubically in some neighborhood of x. Frontini and Sormani [4,5] considered the midpoint rule f (t) dt (x )f ( /2 + x/2) (7) and arrived analogously at a further modified Newton-type iterative scheme f( ) +1 = f ( f( )/(2f ( ))). (8) This scheme has also been derived in [6] by requiring that the iteration function Ψ f (x) = x f(x) f (x + a(x)f(x)) (3) (9)

3 H.H.H. Homeier / Journal of Computational and Applied Mathematics 176 (2005) satisfies Ψ f (x ) = x, Ψ f (x ) = Ψ f (x ) = 0. Hence, a(x ) = [2f (x )] 1 follows. This is satisfied for a(x)= [2f (x)] 1 + b(x)f(x). The scheme (8) is obtained for the special case a(x)= [2f (x)] 1 and using +1 = Φ f ( ). As the modified Newton-type method of Weerakoon and Fernando, scheme (8) converges cubically in some neighborhood of x. Frontini and Sormani [5] also generalized the approach of Weerakoon and Fernando by using general interpolatory quadrature rules of order at least one of the type Q m (f ) = (x ) w j f(η j ), (10) j=1 with η j = +τ j (x ), knots τ j in [0, 1] and weigths w j satisfying m j=1 w j =1 and m j=1 w j τ j =1/2. Thus, in the Newton theorem they approximated f (t) dt Q m (f ), (11) used f(x)= 0 and approximated x in Q m (f ) by x z n = f( )/f ( ) to obtain the iterative schemes +1 = f( ) mj=1 w j f ( η j ) with η j = + τ j (z n ) = τ j f( )/f ( ). All these methods they proved to converge cubically in some neighborhood of x. Now, we introduce the main idea of the present contribution. Instead of using the Newton theorem for y = f(x), we may use it for the inverse function x(y): x(y) = x(y n ) + y y n x (η) dη. Consider an interpolatory quadrature rule R m (f ) = (y y n ) (12) (13) ω γ f(ζ γ ) (14) with ζ γ = y n + τ γ (y y n ), knots τ γ in [0, 1] and weigths ω γ satisfying ω γ = 1, ω γ τ γ = 1/2 (15) such that R m is at least of order 1. Applying it to Eq. (13), we obtain using y = y = y(x ) = 0, x(y)= x, and x(y n ) = equivalent to y n = f( ) the approximate formula x = y n m ω γ x ((1 τ γ )y n ). (16)

4 428 H.H.H. Homeier / Journal of Computational and Applied Mathematics 176 (2005) Since [ 1 x d (y) = dx f(x), x(y)] (17) we have to evaluate the points,γ = x((1 τ γ )y n ). For this, we assume a linear relationship between x and y in the vicinity of x of the form y = f( ) (x )y n with y n = f ( ) whence we obtain,γ = τ γ y n /y n. Putting all together, we obtain a new recursive scheme 1 +1 = S( ); S(x) = x f(x) ω γ f (x τ γ f(x)/f (x)). (18) Note that this scheme is different from all the schemes of form (12) for m>1. To study the convergence order of the newscheme (18), we introduce the notations ε n = x, and C k = f (k) (x )/f (x ). Taylor expansion of G γ (ε n ) = in ε n yields f( )ω γ f ( τ γ f( )/f ( )) (19) G γ (ε n ) = ω γ ε n + 1/2 ω γ C 2 ( τ γ )ε 2 n 1/6 ω γ(2c τ γ C2 2 6τ γc 3 + 3C 3 τ 2 γ 3C2 2 6τ2 γ C2 2 )ε3 n + O(ε4 n ). (20) Summation over γ yields ε n+1 = ε n G γ (ε n ) since = 1 3C C ω γ τ 2 γ (C 3 2C2 2 ) ε 3 n + O(ε 4 n ) (21) ω γ ( 1 + 2τ γ ) = 0 (22) due to Eq. (15). Thus, we have proven the following theorem: Theorem 1. Let ω γ and τ γ for,...,mbe the weights and abscissas, resp., of an m-point interpolatory quadrature for the interval [0, 1], that is at least of order 1. Let f : R R be a smooth function with a simple zero x and abbreviate the scaled derivatives of f at the zero by C k = f (k) (x )/f (x ). Then the iterative scheme +1 = f( ) ω γ 1 f ( τ γ f( )/f ( )) (23)

5 H.H.H. Homeier / Journal of Computational and Applied Mathematics 176 (2005) converges cubically to x in a neighborhood of x. The errors ε n = x obey the order relation ε n+1 = K 3! ε3 n + O(ε4 n ) (24) for ε n 0. Here K is the constant, K = 3C2 2 C ω γ τ 2 γ (C 3 2C2 2 ). (25) We remark that K normally does not vanish even for rules of order greater than 1 for which we have γ ω γτ 2 γ = 1 0 t2 dt = 1/3. Thus for such rules, we have K = C 2 2 (26) which normally does not vanish. The corresponding constant K for the scheme (12) with rules of order greater than 1 is K = 3K/2 (cf. [5, Remark 2]). Specializing to the midpoint rule corresponding to m = 1, ω 1 = 1, and τ 1 = 2 1, we obtain scheme (8) of Frontini, Sormani and Homeier, with ε n+1 = 1 ( 32 6 C C ) 3 ε 3 n + O(ε 4 n ) (27) which is equivalent to the result of Ref. [5]. Specializing to the trapezoidal rule corresponding to m = 2, ω 1 = ω 2 = 2 1, τ 1 = 1 τ 2 = 0, we obtain the newcubically convergent scheme +1 = f(x ( ) n) 1 2 f ( ) + 1 f ( f( )/f (28) ( )) with the order relation ε n+1 = C 3 12 ε3 n + O(ε4 n ). (29) Now, we present some numerical test results for the various cubically convergent schemes, Table 1. All tests were done using MAPLE V TM Release 5 [2,3] on a PC using 32 digit floating point arithmetics (Digits = 32). Compared were the Newton method (N), Eq. (1), the method of Petra and Pták (PP), Eq. (2), the method of Weerakoon and Fernando [8] (WF), Eq. (6), the method of Frontini and Sormani [5], and Homeier [6,7] (FSH), Eq. (8), and the newmethod of Homeier introduced in the present contribution (H), Eq. (28). As test functions, we used the functions f 1 : x x 3 + 4x 2 10, f 2 : x sin(x) 2 x 2 + 1, f 3 : x x 2 exp(x) 3x + 2, (32) f 4 : x cos(x) x, f 5 : x (x 1) 3 1, (30) (31) (33) (34)

6 430 H.H.H. Homeier / Journal of Computational and Applied Mathematics 176 (2005) Table 1 Comparison of various cubically convergent iterative schemes and the Newton method IT NF x f(x ) δx f 1,x 0 = 0.5 N e e 21 PP e e 19 WF e e 29 FSH e e 24 H e e 31 f 1,x 0 = 1 N e e 22 PP e 30 0 WF e e 18 FSH e e 19 H e e 31 f 2,x 0 = 1 N e e 26 PP e e 21 WF e e 30 FSH e 31 0 H e e 21 f 3,x 0 = 3 N e e 26 PP e 20 WF e e 17 FSH e e 25 H e 23 f 4,x 0 = 1 N e e 21 PP e e 32 WF e e 16 FSH e e 32 H e e 32 f 5,x 0 = 2.5 N e 28 PP e 28 WF FSH H e 18 f 6,x 0 = 1.5 N e e 28 PP e 31 0 WF e e 31 FSH e e 31 H e e 23 f 7,x 0 = 2 N e e 21 PP e 31 0 WF e 31 0 FSH e e 23 H e 31 0

7 H.H.H. Homeier / Journal of Computational and Applied Mathematics 176 (2005) Table 1 (continued) IT NF x f(x ) δx f 8,x 0 = 5.55 N e 24 PP WF FSH e 17 H f 8,x 0 = 2.83, IT 60 N e e 20 PP divergent WF e e 28 FSH e 29 H e 24 f 9,x 0 = 0.45, Evaluation via _Dexp N e 25 PP Error, (in evalf/int) unable to handle singularity WF FSH H e e 19 f 9,x 0 = 0.45, Evaluation via _NCrule N e 17 PP Error, (in evalf/int) argument too large WF FSH H f 9,x 0 = 2.45, Evaluation via _NCrule N e 17 PP e 17 WF FSH H e 17 f 6 : x x 3 10, (35) f 7 : x x exp(x 2 ) sin(x) cos(x) + 5, (36) f 8 : x exp(x 2 + 7x 30) 1, (37) f 9 : x 0 (exp( t 3 /2) exp( t 8 /2)) dt + 1/10. (38) For all functions except f 9 defined in Eq. (38), the number of functional evaluations (NF) is counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative. For the function f 9, NF is given by the number of evaluations of the integrand as called using numerical quadrature, i.e., the derivative. For this function, it is also studied whether the numerical quadrature scheme influences the performance of the various iterative schemes. Two automatic quadrature schemes are compared that are available in Maple: _Dexp, a double exponential method, and _NCrule

8 432 H.H.H. Homeier / Journal of Computational and Applied Mathematics 176 (2005) that is based on Newton Cotes rules. As convergence criterion, it was required that the distance of two consecutive approximations δx for the zero was less than Also displayed are the number of iterations (IT), the approximate zero x, and the value f(x ) at this position. The results showthat the cubically convergent schemes, especially the newmethod of Homeier (H), Eq. (28) can compete with the Newton method and seem to be superior in difficult cases and in cases involving numerical quadrature as exemplified in function f 9 defined in Eq. (37). References [1] S. Amat, S. Busquier, J.M. Gutiérrez, Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. Appl. Math. 157 (2003) [2] B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, S.M. Watt, Maple V Language Reference Manual, Springer, Berlin, [3] B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, S.M. Watt, Maple V Library Reference Manual, Springer, Berlin, [4] M. Frontini, E. Sormani, Modified Newton s method with third-order convergence and multiple roots, J. Comput. Appl. Math. 156 (2003) [5] M. Frontini, E. Sormani, Some variant of Newton s method with third-order convergence, J. Comput. Appl. Math. 140 (2003) [6] H.H.H. Homeier, A modified Newton method for root finding with cubic convergence, J. Comput. Appl. Math. 157 (2003) [7] H.H.H. Homeier, A modified Newton method with cubic convergence: the multivariate case, J. Comput. Appl. Math. 169 (2004) [8] S. Weerakoon, T.G.I. Fernando, A variant of Newton s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000)

A modied Newton method with cubic convergence: the multivariate case

Journal of Computational and Applied Mathematics 169 (2004) 161 169 www.elsevier.com/locate/cam A modied Newton method with cubic convergence: the multivariate case H.H.H. Homeier a;b; a Institut fur Physikalische